In the cases where \(\sum _{k=1}^p w_k^{S} {1}(c\in T_{ik}^S) = 0.5* \sum _{k=1}^p w_k^{S} \) , we propose to find the maximum number of compatible clusters to add into the aggregated tree \(Clu(T_{i}^{S^*})\) . Although adding these clusters into the constructed aggregation tree does not affect the resulting total \(RF\)  distance, we favor the aggregated trees with more compatible clusters since they contain as many details from the input trees. We conduct the following steps. First we form a set \(C^{\prime }_i\)  that includes all clusters such that \(\sum _{k=1}^p w_k^{S} {1}(c\in T_{ik}^S) = 0.5* \sum _{k=1}^p w_k^{S} \) . Then we construct the incompatibility graph by treating the clusters as nodes and adding an edge if two clusters are not compatible. Finding the maximum number of compatible clusters is then equivalent to the maximum independent set problem [1]}. This strong NP-hard problem can be addressed by the existing methods [2]}, [3]}.
